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Essays on Econometric Modelling of Temporal Networks Matteo Iacopini

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Essays on Econometric Modelling of Temporal Networks Matteo Iacopini Essays on econometric modelling of temporal networks Matteo Iacopini To cite this version: Matteo Iacopini. Essays on econometric modelling of temporal networks. Statistical Finance [q-fin.ST]. Université Panthéon-Sorbonne - Paris I, 2018. English. NNT : 2018PA01E058. tel-02175727 HAL Id: tel-02175727 https://tel.archives-ouvertes.fr/tel-02175727 Submitted on 6 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Corso di Dottorato di ricerca in Economia Doctorat en Mathématiques Appliquées cotutela con Université Paris I - Panthéon-Sorbonne Ciclo XXX Tesi di Ricerca Essays on the econometric modelling of temporal networks SSD: SECS-P/05 Coordinatore del Dottorato prof. Giacomo Pasini Primo Supervisore prof. Monica Billio Secondo Supervisore prof. Dominique Guégan Terzo Supervisore prof. Roberto Casarin Dottorando Matteo Iacopini Matricola 956154 CA’FOSCARI UNIVERSITY OF VENICE AND UNIVERSITÉ PARIS IPANTHÉON-SORBONNE DOCTORAL THESIS Essays on the econometric modelling of temporal networks Author: Supervisors: Matteo IACOPINI Prof. Monica BILLIO Prof. Dominique GUÉGAN Prof. Roberto CASARIN A thesis submitted in fulfilment of the requirements for the degree of PhD in Economics Doctorat en Mathématiques Appliquées in the Department of Economics and Centre d’Économie de la Sorbonne July 5, 2018 iii Declaration of Authorship I, Matteo IACOPINI, declare that this thesis titled, “Essays on the econometric modelling of temporal networks” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research degree at these Universities. • Where any part of this thesis has previously been submitted for a degree or any other qualification at these Universities or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: v “God used beautiful mathematics in creating the world.” Paul Dirac “Mathematics is the language in which God has written the universe.” Galileo Galilei “The most important questions of life are, for the most part, really only problems of probability.” Pierre Simon, Marquis de Laplace vii CA’ FOSCARI UNIVERSITY OF VENICE AND UNIVERSITÉ PARIS I PANTHÉON-SORBONNE Abstract Department of Economics PhD in Economics Doctorat en Mathématiques Appliquées Essays on the econometric modelling of temporal networks by Matteo IACOPINI Graph theory has long been studied in mathematics and probability as a tool for describing dependence between nodes. However, only recently it has been implemented on data, givin birth to the statistical analysis of real networks. The topology of economic and financial networks is remarkably complex: it is generally unobserved, thus requiring adequate inferential procedures for it estimation, moreover not only the nodes, but the structure of dependence itself evolves over time. Statistical and econometric tools for modelling the dynamics of change of the network structure are lacking, despite their increasing requirement in several fields of research. At the same time, with the beginning of the era of “Big data” the size of available datasets is becoming increasingly high and their internal structure is growing in complexity, hampering traditional inferential processes in multiple cases. This thesis aims at contributing to this newborn field of literature which joins probability, economics, physics and sociology by proposing novel statistical and econometric method- ologies for the study of the temporal evolution of network structures of medium-high di- mension. ix CA’ FOSCARI UNIVERSITY OF VENICE AND UNIVERSITÉ PARIS I PANTHÉON-SORBONNE Abstract Department of Economics PhD in Economics Doctorat en Mathématiques Appliquées Essays on the econometric modelling of temporal networks by Matteo IACOPINI La théorie des graphes a longtemps été étudiée en mathématiques et en probabilité en tant qu’outil pour décrire la dépendance entre les nœuds. Cependant, ce n’est que récemment qu’elle a été mise en œuvre sur des données, donnant naissance à l’analyse statistique des réseaux réels. La topologie des réseaux économiques et financiers est remarquablement complexe: elle n’est généralement pas observée, et elle nécessite ainsi des procédures inférentielles adéquates pour son estimation, d’ailleurs non seulement les nœuds, mais la structure de la dépendance elle-même évolue dans le temps. Des outils statistiques et économétriques pour modéliser la dynamique de changement de la structure du réseau font défaut, malgré leurs besoins crois- sants dans plusieurs domaines de recherche. En même temps, avec le début de l’ère des “Big data”, la taille des ensembles de données disponibles devient de plus en plus élevée et leur structure interne devient de plus en plus complexe, entravant les processus inférentiels traditionnels dans plusieurs cas. Cette thèse a pour but de contribuer à ce nouveau champ littéraire qui associe proba- bilités, économie, physique et sociologie en proposant de nouvelles méthodologies statis- tiques et économétriques pour l’étude de l’évolution temporelle des structures en réseau de moyenne et haute dimension. xi Contents Declaration of Authorship iii Abstract vii Abstract ix Contents xi List of Figures xv List of Tables xxi List of Symbols xxi xxiii 1 Introduction 1 1.1 Preliminaries ...................................... 1 1.1.1 Networks ................................... 1 1.1.2 Temporal and multi-layer networks .................... 4 1.1.3 Multi-dimensional datasets ......................... 6 1.2 Motivation ....................................... 10 1.2.1 Dynamic networks .............................. 10 1.2.2 High-dimensionality ............................. 11 1.3 Contribution ...................................... 12 1.4 Outline of the thesis .................................. 12 2 Bayesian Dynamic Tensor Regression 15 2.1 Introduction ...................................... 15 2.2 A Tensor Regression Model ............................. 17 2.2.1 Tensor Calculus and Decompositions ................... 17 2.2.2 A General Dynamic Model ......................... 20 2.2.3 Important special cases ........................... 22 2.3 Bayesian Inference .................................. 23 2.3.1 Prior Specification .............................. 24 2.3.2 Posterior Computation ............................ 26 2.4 Simulation Results .................................. 28 2.5 Application ...................................... 32 2.5.1 Impulse response analysis .......................... 34 2.6 Conclusions ...................................... 37 3 Bayesian Markov Switching Tensor Regression for Time-varying Networks 39 3.1 Introduction ...................................... 39 3.2 A Markov switching model for networks ..................... 40 3.3 Bayesian Inference .................................. 43 3.4 Posterior Approximation ............................... 47 3.5 Simulation Results .................................. 49 xii 3.6 Applications ...................................... 52 3.6.1 Data description ............................... 52 3.6.2 Results ..................................... 55 3.7 Conclusions ...................................... 58 4 Nonparametric forecasting of multivariate probability density functions 61 4.1 Introduction ...................................... 61 4.2 Preliminaries ...................................... 66 4.2.1 Notation .................................... 66 4.2.2 Related literature ............................... 68 4.3 Methodology ..................................... 70 4.3.1 Step 1 - Copula estimation .......................... 71 4.3.2 Step 2 - Modified fPCA ........................... 73 4.3.3 Step 3 - Prediction .............................. 76 4.4 Extensions ....................................... 77 4.4.1 Unbounded support ............................. 77 4.4.2 Multivariate case: d > 2 ........................... 79 4.5 Application ...................................... 79 4.6 Conclusions ...................................... 84 5 Conclusions 85 A Appendix A 87 A.1 Background material on tensor calculus ...................... 87 B Appendix B 95 B.1 Proofs of the results in Section 2.2 .......................... 95 B.2 Initialisation details .................................. 97 B.3 Computational details - matrix case ........................ 98 B.3.1 Full conditional distribution of φ ...................... 98 B.3.2 Full conditional distribution of τ ...................... 99 B.3.3 Full conditional distribution of λj,r ..................... 100 B.3.4 Full conditional distribution of wj,r,p .................... 101 (r) B.3.5 Full conditional distribution
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